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Mat. Zametki, 1972, Volume 12, Issue 6, Pages 665–670 (Mi mz9931)  

A note on a theorem of Sunouchi

A. V. Efimov

Moscow Institute of Electronic Technology

Abstract: We show that for negative $\alpha$ Sunouchi's formula
\begin{gather*} H_n(f,\alpha,\beta,x)=\frac1{A^\beta_n}\sum_{k=0}^nA_{n-k}^{\beta-1}|f(x)-\sigma_k^\alpha(f,x)|,
\alpha>-\frac12,\quad\beta>\frac12, \end{gather*}
becomes false, where $\sigma_k^\alpha(f,x)$ is the $(C,\alpha)$ mean of the Fourier series for the function $f(x)\in\mathrm{Lip} \gamma$, $0<\gamma<1$. A bound is given for $H_n(f,\alpha,\beta,x)$ for all $\alpha>-1$, $\beta>-1$, which for $\alpha+\beta>0$, $\alpha\geqslant0$, $\beta\geqslant0$, coincides with the Sunouchi bound. The proof is by a method different from that of Sunouchi.

Full text: PDF file (379 kB)

English version:
Mathematical Notes, 1972, 12:6, 839–842

Bibliographic databases:

UDC: 517.5
Received: 05.01.1972

Citation: A. V. Efimov, “A note on a theorem of Sunouchi”, Mat. Zametki, 12:6 (1972), 665–670; Math. Notes, 12:6 (1972), 839–842

Citation in format AMSBIB
\Bibitem{Efi72}
\by A.~V.~Efimov
\paper A note on a theorem of Sunouchi
\jour Mat. Zametki
\yr 1972
\vol 12
\issue 6
\pages 665--670
\mathnet{http://mi.mathnet.ru/mz9931}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=324291}
\zmath{https://zbmath.org/?q=an:0247.42007}
\transl
\jour Math. Notes
\yr 1972
\vol 12
\issue 6
\pages 839--842
\crossref{https://doi.org/10.1007/BF01156041}


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